Statistical Mechanics · Problem 9.12

Statistical Mechanics of Interacting Systems: The Method of Quantized Fields — R. K. Pathria

Physics — Statistical Mechanics · boltzmannian-gas-second-virial-coefficient

Problem

Show that the coefficient

\mathfrak{b}_2

for a quantum-mechanical Boltzmannian gas composed of ``spinless'' particles satisfies the following relations:

\mathfrak{b}_2 = \lim_{J\to\infty}\left\{\frac{1}{(2J+1)^2}\mathfrak{b}_2^S(J)\right\} = \lim_{J\to\infty}\left\{\frac{1}{(2J+1)^2}\mathfrak{b}_2^A(J)\right\}

= \tfrac{1}{2}[\mathfrak{b}_2^S(0) + \mathfrak{b}_2^A(0)].

Obtain the value of

\mathfrak{b}_2

, to fifth order in

(r_0/\lambda)

, by using the Beth–Uhlenbeck expressions (9.5.36, 37), and compare your result with the classical value of

\mathfrak{b}_2

, namely

-(2\pi/3)(r_0/\lambda)^3